\subsection{Shizhong's comment}

Shizhong suggested to set $\eta=0$, therefore the $v^2+w^2$ in kinetic energy can be expressed as simple form of $F^o, F^c$. And we change T-matrix, (probably only the open-open element of $2\times2$ T-matrix), to sweep the crossover region.  This way the algebra might be simple significantly.  \emph{Very smart!}

Anderson has a paper about why the mean field approach of BCS is so successful.  The plasma mode might suppress the excitation of finite momentum pair.  (name is something about Collection mode RPA...)

Larkin has a book about fluctuation of BCS.  Theory of Fluctuation in Superconductor.  
